Introduction to Linear Regression

Relationships Between Variables

  • In a statistical model, we generally have one variable that is the output and one or more variables that are the inputs.
  • Response variable
    • a.k.a. \(y\), dependent
    • The quantity you want to understand
    • In this class – always numerical
  • Explanatory variable
    • a.k.a. \(x\), independent, explanatory, predictor
    • Something you think might be related to the response
    • Either numerical or categorical

Visualizing Linear Regression


  • The scatterplot has been called the most “generally useful invention in the history of statistical graphics.”
  • It is a simple two-dimensional plot in which the two coordinates of each dot represent the values of two variables measured on a single observation.

Characterizing Relationships


  • Form (e.g. linear, quadratic, non-linear)

  • Direction (e.g. positive, negative)

  • Strength (how much scatter/noise?)

  • Unusual observations (do points not fit the overall pattern?)

Data for Today

The ncbirths dataset is a random sample of 1,000 cases taken from a larger dataset collected in North Carolina in 2004.

Each case describes the birth of a single child born in North Carolina, along with various characteristics of the child (e.g. birth weight, length of gestation, etc.), the child’s mother (e.g. age, weight gained during pregnancy, smoking habits, etc.) and the child’s father (e.g. age).

Your Turn!

How would your characterize this relationship?

  • shape
  • direction
  • strength
  • outliers

What if you added another variable?

Summarizing a Linear Relationship


Correlation:

strength and direction of a linear relationship between two quantitative variables

  • Correlation coefficient between -1 and 1
  • Sign of the correlations shows direction
  • Magnitude of the correlation shows strength

Anscombe Correlations

Four datasets, very different graphical presentations

  • same mean and standard deviation in both \(x\) and \(y\)
  • same correlation
  • same regression line

For which of these relationships is correlation a reasonable summary measure?

Calculating Correlation in R

births_post26 <- ncbirths %>% 
  drop_na(weight, weeks) %>% 
  filter(weeks > 26)


get_correlation(births_post26, 
                weeks ~ weight)
# A tibble: 1 × 1
    cor
  <dbl>
1 0.578

The Importance of Language

  • The word “correlation” has both a precise mathematical definition and a more general definition for typical usage in English.

  • These uses are obviously related and generally in sync.

  • There are times when these two uses can be conflated and/or misconstrued.


Linear regression:

we assume the the relationship between our response variable (\(y\)) and explanatory variable (\(x\)) can be modeled with a linear function, plus some random noise

\(response = intercept + slope \cdot explanatory + noise\)

Writing the Regression Equation


Population Model

\(y = \beta_0 + \beta_1 \cdot x + \epsilon\)


\(y\) = response

\(\beta_0\) = population intercept

\(\beta_1\) = population slope

\(\epsilon\) = errors / residuals

Sample Model

\(\widehat{y} = b_0 + b_1 \cdot x\)


Why does this equation have a hat on \(y\)?

Linear Regression with One Numerical Explanatory Variable

Obtaining Coefficient Estimates

Step 1: Fit a linear regression

weeks_lm <- lm(weight ~ weeks, data = births_post26)


Step 2: Obtain coefficient table

get_regression_table(weeks_lm)
# A tibble: 2 × 7
  term      estimate std_error statistic p_value lower_ci upper_ci
  <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept   -5.34      0.565     -9.45       0   -6.45    -4.23 
2 weeks        0.325     0.015     22.2        0    0.296    0.354

Our focus (for now…)

Estimated regression equation

\[\widehat{y} = b_0 + b_1 \cdot x\]

# A tibble: 2 × 7
  term      estimate std_error statistic p_value lower_ci upper_ci
  <chr>        <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept   -5.34      0.565     -9.45       0   -6.45    -4.23 
2 weeks        0.325     0.015     22.2        0    0.296    0.354


Write out the estimated regression equation!

How do you interpret the intercept value of -5.341?

How do you interpret the slope value of 0.325?

Obtaining Residuals


\(\widehat{weight} = -5.341+0.325 \cdot weeks\)


What would the residual be for a pregnancy that lasted 39 weeks and whose baby weighed 7.63 pounds?

Linear Regression with One Categorical Explanatory Variables

Step 1: Finding distinct levels

distinct(births_post26, habit)
# A tibble: 2 × 1
  habit    
  <fct>    
1 nonsmoker
2 smoker   

Step 2: Fit a linear regression

habit_lm <- lm(weight ~ habit, data = births_post26)


Step 3: Obtain coefficient table

get_regression_table(habit_lm)
# A tibble: 2 × 7
  term          estimate std_error statistic p_value lower_ci upper_ci
  <chr>            <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept         7.23     0.047    155.     0        7.14     7.32 
2 habit: smoker    -0.4      0.13      -3.07   0.002   -0.656   -0.145

🤔

Estimated Regression Equation

# A tibble: 2 × 7
  term          estimate std_error statistic p_value lower_ci upper_ci
  <chr>            <dbl>     <dbl>     <dbl>   <dbl>    <dbl>    <dbl>
1 intercept         7.23     0.047    155.     0        7.14     7.32 
2 habit: smoker    -0.4      0.13      -3.07   0.002   -0.656   -0.145


\[\widehat{weight} = 7.23 - 0.4 \cdot Smoker\]


But what does \(Smoker\) represent???

Indicator Variables

\[ \widehat{y} = b_0 + b_1 \cdot x \]

\(x\) is a categorical variable with levels:

  • "nonsmoker"
  • "smoker"

We need to convert to:

  • a “baseline” group
  • “offsets” / adjustments to the baseline

Based on the regression table, what habit group was chosen to be the baseline?

A More Intuitive Equation

\[\widehat{weight} = 7.23 - 0.4 \cdot 1_{Smoker}(x)\]

where

\(1_{smoker}(x) = 1\) if the mother was a "smoker"

\(1_{smoker}(x) = 0\) if the mother was a "nonsmoker"

Obtaining Group Means

\[\widehat{weight} = 7.23 - 0.4 \cdot 1_{Smoker}(x)\]

Given the equation, what is the estimated mean birth weight for nonsmoking mothers?


For smoking mothers?

Choose Your Adventure

Statistical Critique

Midterm Project Proposal